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Flat knot 6.62

Min(phi) over symmetries of the knot is: [-4,-4,0,1,3,4,0,1,3,2,4,2,4,3,5,1,1,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.62']
Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.62', '6.176', '6.181', '6.194', '6.228', '6.267', '6.268', '6.449']
Outer characteristic polynomial of the knot is: t^7+153t^5+206t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.62']
2-strand cable arrow polynomial of the knot is: -144*K1**4 + 32*K1**3*K2*K3 - 192*K1**3*K3 - 480*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 32*K1**2*K2*K4 + 2776*K1**2*K2 - 176*K1**2*K3**2 - 3432*K1**2 - 1056*K1*K2**2*K3 - 448*K1*K2*K3*K4 + 3784*K1*K2*K3 + 1256*K1*K3*K4 + 144*K1*K4*K5 + 24*K1*K5*K6 - 72*K2**4 - 96*K2**3*K6 + 96*K2**2*K3**2*K4 - 896*K2**2*K3**2 - 64*K2**2*K3*K7 - 8*K2**2*K4**2 + 1232*K2**2*K4 - 72*K2**2*K6**2 - 3378*K2**2 - 32*K2*K3**2*K4 + 976*K2*K3*K5 + 320*K2*K4*K6 + 24*K2*K5*K7 + 64*K2*K6*K8 - 64*K3**2*K4**2 + 32*K3**2*K6 - 1944*K3**2 + 32*K3*K4*K7 + 8*K3*K5*K8 - 932*K4**2 - 248*K5**2 - 150*K6**2 - 8*K7**2 - 18*K8**2 + 3108
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.62']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81586', 'vk6.81666', 'vk6.81668', 'vk6.81903', 'vk6.81909', 'vk6.82101', 'vk6.82268', 'vk6.82272', 'vk6.82342', 'vk6.82346', 'vk6.82620', 'vk6.82624', 'vk6.82862', 'vk6.82878', 'vk6.83146', 'vk6.83148', 'vk6.83376', 'vk6.83390', 'vk6.84158', 'vk6.84654', 'vk6.84964', 'vk6.84972', 'vk6.85956', 'vk6.85958', 'vk6.86175', 'vk6.86191', 'vk6.86425', 'vk6.88128', 'vk6.89048', 'vk6.89052', 'vk6.89722', 'vk6.90038']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4O5O6U2U1U4U6U3U5
R3 orbit {'O1O2O3O4O5O6U2U1U4U6U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U4U1U3U6U5
Gauss code of K* O1O2O3O4O5O6U2U1U5U3U6U4
Gauss code of -K* O1O2O3O4O5O6U3U1U4U2U6U5
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -4 -4 1 0 4 3],[ 4 0 0 4 2 5 3],[ 4 0 0 3 1 4 2],[-1 -4 -3 0 -1 2 1],[ 0 -2 -1 1 0 2 1],[-4 -5 -4 -2 -2 0 0],[-3 -3 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 4 3 1 0 -4 -4],[-4 0 0 -2 -2 -4 -5],[-3 0 0 -1 -1 -2 -3],[-1 2 1 0 -1 -3 -4],[ 0 2 1 1 0 -1 -2],[ 4 4 2 3 1 0 0],[ 4 5 3 4 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,-1,0,4,4,0,2,2,4,5,1,1,2,3,1,3,4,1,2,0]
Phi over symmetry [-4,-4,0,1,3,4,0,1,3,2,4,2,4,3,5,1,1,2,1,2,0]
Phi of -K [-4,-4,0,1,3,4,0,2,1,4,3,3,2,5,4,0,2,2,1,1,1]
Phi of K* [-4,-3,-1,0,4,4,1,1,2,3,4,1,2,4,5,0,1,2,2,3,0]
Phi of -K* [-4,-4,0,1,3,4,0,1,3,2,4,2,4,3,5,1,1,2,1,2,0]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+95t^4+24t^2
Outer characteristic polynomial t^7+153t^5+206t^3+4t
Flat arrow polynomial -2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -144*K1**4 + 32*K1**3*K2*K3 - 192*K1**3*K3 - 480*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 32*K1**2*K2*K4 + 2776*K1**2*K2 - 176*K1**2*K3**2 - 3432*K1**2 - 1056*K1*K2**2*K3 - 448*K1*K2*K3*K4 + 3784*K1*K2*K3 + 1256*K1*K3*K4 + 144*K1*K4*K5 + 24*K1*K5*K6 - 72*K2**4 - 96*K2**3*K6 + 96*K2**2*K3**2*K4 - 896*K2**2*K3**2 - 64*K2**2*K3*K7 - 8*K2**2*K4**2 + 1232*K2**2*K4 - 72*K2**2*K6**2 - 3378*K2**2 - 32*K2*K3**2*K4 + 976*K2*K3*K5 + 320*K2*K4*K6 + 24*K2*K5*K7 + 64*K2*K6*K8 - 64*K3**2*K4**2 + 32*K3**2*K6 - 1944*K3**2 + 32*K3*K4*K7 + 8*K3*K5*K8 - 932*K4**2 - 248*K5**2 - 150*K6**2 - 8*K7**2 - 18*K8**2 + 3108
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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